8. Definite Integrals

Identifying Riemann sums Fill in the blanks with an interval and a value of n.

4

∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with

k = 1

n = ________ .

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₀⁴ (4𝓍― 𝓍²) d𝓍

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,5] with 5 subintervals.

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = 1/x on [1,6] ; n = 5

(d) Calculate the midpoint Riemann sum.

15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.

15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals

Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.

ƒ(𝓍) = x + 1 on [1,6] ; n = 5

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

ƒ(x) = 4 - 2x on [0,4]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the 𝓍-axis by dividing the interval [1, 7] into n = 6 subintervals. Use a left and right Riemann sum to obtain two different approximations.                                                                                                                                                                         

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(b) 4 + 5 + 6 + 7 + 8 + 9

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

II. Using the Trapezoidal Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.

∫ from 1 to 2 of 1 / s² ds

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 3 of 1/√(x + 1) dx

Approximate the area under the curve $f\left(x\right)=\sqrt{x+3}$ over the interval $[1,5]$ using the Right Riemann sum with 8 subintervals.